Anick's spaces and the double loops of odd primary Moore spaces

Author:
Stephen D. Theriault

Journal:
Trans. Amer. Math. Soc. **353** (2001), 1551-1566

MSC (2000):
Primary 55P35; Secondary 55Q25

Published electronically:
October 11, 2000

MathSciNet review:
1709779

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Several properties of Anick's spaces are established which give a retraction of Anick's off if and . The proof is alternate to and more immediate than the two proofs of Neisendorfer's.

**[A]**David Anick,*Differential algebras in topology*, Research Notes in Mathematics, vol. 3, A K Peters, Ltd., Wellesley, MA, 1993. MR**1213682****[AG]**David Anick and Brayton Gray,*Small 𝐻 spaces related to Moore spaces*, Topology**34**(1995), no. 4, 859–881. MR**1362790**, 10.1016/0040-9383(95)00001-1**[CM]**F. R. Cohen and M. E. Mahowald,*A remark on the self-maps of Ω²𝑆²ⁿ⁺¹*, Indiana Univ. Math. J.**30**(1981), no. 4, 583–588. MR**620268**, 10.1512/iumj.1981.30.30046**[CMN1]**F. R. Cohen, J. C. Moore, and J. A. Neisendorfer,*Torsion in homotopy groups*, Ann. of Math. (2)**109**(1979), no. 1, 121–168. MR**519355**, 10.2307/1971269**[CMN2]**F. R. Cohen, J. C. Moore, and J. A. Neisendorfer,*The double suspension and exponents of the homotopy groups of spheres*, Ann. of Math. (2)**110**(1979), no. 3, 549–565. MR**554384**, 10.2307/1971238**[CMN3]**F. R. Cohen, J. C. Moore, and J. A. Neisendorfer,*Exponents in homotopy theory*, Algebraic topology and algebraic 𝐾-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 3–34. MR**921471****[G1]**Brayton Gray,*Homotopy commutativity and the 𝐸𝐻𝑃 sequence*, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 181–188. MR**1022680**, 10.1090/conm/096/1022680**[G2]**Brayton Gray,*On the iterated suspension*, Topology**27**(1988), no. 3, 301–310. MR**963632**, 10.1016/0040-9383(88)90011-0**[G3]**Brayton Gray,*𝐸𝐻𝑃 spectra and periodicity. I. Geometric constructions*, Trans. Amer. Math. Soc.**340**(1993), no. 2, 595–616. MR**1152323**, 10.1090/S0002-9947-1993-1152323-X**[J]**I. M. James,*Reduced product spaces*, Ann. of Math. (2)**62**(1955), 170–197. MR**0073181****[N1]**Joseph Neisendorfer,*Primary homotopy theory*, Mem. Amer. Math. Soc.**25**(1980), no. 232, iv+67. MR**567801**, 10.1090/memo/0232**[N2]**Joseph Neisendorfer,*Product decompositions of the double loops on odd primary Moore spaces*, Topology**38**(1999), no. 6, 1293–1311. MR**1690159**, 10.1016/S0040-9383(98)00055-X**[N3]**J. A. Neisendorfer,*James-Hopf invariants, Anick's spaces, and the double loops on odd primary Moore spaces*, Canad. Math. Bull.**43**(2000), 226-235. CMP**2000:11****[S1]**Paul Selick,*Odd primary torsion in 𝜋_{𝑘}(𝑆³)*, Topology**17**(1978), no. 4, 407–412. MR**516219**, 10.1016/0040-9383(78)90007-1**[S2]**Paul Selick,*A decomposition of 𝜋_{∗}(𝑆^{2𝑝+1};𝑍/𝑝𝑍)*, Topology**20**(1981), no. 2, 175–177. MR**605656**, 10.1016/0040-9383(81)90036-7**[S3]**Paul Selick,*A reformulation of the Arf invariant one mod 𝑝 problem and applications to atomic spaces*, Pacific J. Math.**108**(1983), no. 2, 431–450. MR**713746****[T1]**S. D. Theriault,*A reconstruction of Anick's fibration*, to appear in Topology.**[T2]**S. D. Theriault,*Properties of Anick's spaces*, to appear in Trans. Amer. Math. Soc. CMP**2000:01**

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Additional Information

**Stephen D. Theriault**

Affiliation:
Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607

Address at time of publication:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Email:
st7b@virginia.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-00-02622-2

Keywords:
Loop space decomposition,
Hopf invariant

Received by editor(s):
December 1, 1998

Published electronically:
October 11, 2000

Additional Notes:
The author was supported in part by an NSERC Postdoctoral Fellowship

Article copyright:
© Copyright 2000
American Mathematical Society